Donnarumma collected a back pass from Alessio Romagnoli but gave the ball straight to Sampdoria forward Gregoire Defrel who tapped in from close range after 33 seconds to hand Milan a second successive league defeat.
Milan gave a disjointed performance and often struggled to play their way out of defence as eighth-placed Sampdoria pressed aggressively.
Sampdoria striker Fabio Quagliarella missed two first-half chances and the 36-year-old, Serie A's leading scorer, crashed another effort against the crossbar after the break.
Fourth-placed Milan (51 points) never gave up, however, and had a late penalty appeal turned down following a long VAR review of a tackle by Nicola Murru on Krzysztof Piatek.
(Reporting by Brian Homewood, editing by Ed Osmond)
